Strong convergence for tensor GUE random matrices

TL;DR

This paper extends the strong convergence results of GUE matrices to multipartite quantum systems, showing that under certain conditions, strong asymptotic freeness persists in complex quantum models.

Contribution

It introduces conditions under which strong asymptotic freeness holds for GUE matrices acting on multipartite quantum systems, expanding previous results to more complex settings.

Findings

Strong asymptotic freeness in multipartite systems

Conditions on site dimensions for convergence

Application of interpolation technology in proof

Abstract

Haagerup and Thorbj{\o}rnsen proved that iid GUEs converge strongly to free semicircular elements as the dimension grows to infinity. Motivated by considerations from quantum physics -- in particular, understanding nearest neighbor interactions in quantum spin systems -- we consider iid GUE acting on multipartite state spaces, with a mixing component on some sites and identity on the remaining sites. We show that under proper assumptions on the dimension of the sites, strong asymptotic freeness still holds. Our proof relies on an interpolation technology recently introduced by Bandeira, Boedihardjo and van Handel.